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Recognizing divisibility

Recognizing Divisibility. Created by Sal Khan and Monterey Institute for Technology and Education.

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Video transcript

Determine whether 380 is divisible by 2, 3, 4, 5, 6, 9 or 10. They skipped 7 and 8 so we don't have to worry about those. So let's think about 2. So are we divisible by 2? Let me write the 2 here. Well, in order for something to be divisible by 2, it has to be an even number, and to be an even number, your ones digit-- so let me rewrite 380. To be even, your ones digit has to be even, so this has to be even. And for this to be even, it has to be 0, 2, 4, 6 or 8, and this is 0, so 380 is even, which means it is divisible by 2, so it works with 2. So 2 works out. Let's think about the situation for 3. Now, a quick way to think about 3-- so let me write just 3 question mark-- is to add the digits of your number. And if the sum that you get is divisible by 3, then you are divisible by 3. So let's try to do that. So 380, let's add the digits. 3 plus 8 plus 0 is equal to-- 3 plus 8 is 11 plus 0, so it's just 11. And if you have trouble figuring out whether this is divisible by 3, you could then just add these two numbers again, so you can actually add the 1 plus 1 again, and you would get a 2. Regardless of whether you look at the 1 or the 2, neither of these are divisible by 3. So not divisible by 3, and maybe in a future video, I'll explain why this works, and maybe you want to think about why this works. So these aren't divisible by 3, so 380 is not divisible. 380, not divisible by 3, so 3 does not work. We are not divisible by 3. Now, I'll think about the situation for 4, so we're thinking about 4 divisibility. So let me write it in orange. So we are wondering about 4. Now, something you may or may not already realize is that 100 is divisible by 4. It goes evenly. So this is 380. So the 300 is divisible by 4, so we just have to figure out whether the leftover, whether the 80, is divisible by 4. Another way to think about it is are the last two digits divisible by 4? And this comes from the fact that 100 is divisible by 4, so everything, the hundreds place or above, it's going to be divisible by 4. You just have to worry about the last part. So in this situation, is 80 divisible by 4? Now, you could eyeball that. You could say, well, 8 is definitely divisible by 4. 8 divided by 4 is 2. 80 divided by 4 is 20, so this works. Yes! Yes! So since 80 is divisible by 4, 380 is also divisible by 4, so 4 works. So let's do 5. I'll actually scroll down a little bit. Let's try 5. So what's the pattern when something is divisible by 5? Let's do the multiple of 5? 5, 10, 15, 20, 25. So if something's divisible by 5-- I could keep going-- that means it ends with either a 5 or a 0, right? Every multiple of 5 either has a 5 or a 0 in the ones place. Now 380 has a 0 in the ones place, so it is divisible by 5. Now, let's think about the situation for 6. Let's think about what happens with 6. So we want to know are we divisible by 6? So to be divisible by 6, you have to be divisible by the things that make up 6. Remember, 6 is equal to 2 times 3. So if you're divisible by 6, that means you are divisible by 2 and you are divisible by 3. If you're divisible by both 2 and 3, you'll be divisible by 6. Now, 380 is divisible by 2, but we've already established that it is not divisible by 3. If it's not divisible by 3, it cannot be divisible by 6, so this gets knocked out. We are not divisible by 6. Now, let's go to 9. So divisibility by 9. So you can make a similar argument here that if something is not divisible by 3, there's no way it's going to be divisible by 9 because 9 is equal to 3 times 3. So to be divisible by 9, you have to be divisible by 3 at least twice. At least two 3's have to go into your number, and this isn't the case, so you could already knock 9 out. But if we didn't already know that we're not divisible by 3, the other way to do it is a very similar way to figure out divisibility by 3. We can add the digits. So you add 3 plus 8 plus 0, and you get 11. And you say is this divisible by 9? And you say this is not divisible by 9, so 380 must not be divisible by 9. And for 3, you do the same thing, but you test whether the sum is divisible by 3. For 9, you test whether it's divisible by 9. So lastly, we have the number 10. We have the number 10, and this is on some level the easiest one. What do all the multiples of 10 look like? 10, 20, 30, 40, we could just keep going on and on. They all end with zero. Or if something ends with zero, it is divisible by 10. 380 does end with zero, or its ones place does have a zero on it, so it is divisible by 10. So we're divisible by all of these numbers except for 3, 6 and 9.